Estimation of arc location in three dimensions

ABSTRACT

Multiple magnetic field sensors are arranged around a current-containing volume at multiple longitudinal and circumferential positions. Each sensor measures multiple magnetic field components and is characterized by one or more calibration parameters. A longitudinal primary current flows through two end-to-end electrical conductors that are separated by an arc gap, and flows as at least one longitudinal primary electric arc that spans the arc gap and that moves transversely within the arc gap. Estimated transverse position of the primary electric arc is calculated, based on the longitudinal position of the arc gap, and two or more of the measured magnetic field components along with one or more corresponding sensor positions or calibration parameters. In addition, estimated occurrence, position, and magnitude of a transverse secondary current (i.e., a side arc) can be calculated based on those quantities.

BENEFIT CLAIM TO RELATED APPLICATION

This application is a continuation of U.S. non-provisional applicationSer. No. 16/719,792 entitled “Estimation of arc location in threedimensions” filed Dec. 18, 2019 in the names Matthew A. Cibula, Paul E.King, and C. Rigel Woodside (now U.S. Pat. No. 11,022,656), which is acontinuation of U.S. non-provisional application Ser. No. 15/715,018entitled “Estimation of arc location in three dimensions” filed Sep. 25,2017 in the names Matthew A. Cibula, Paul E. King, and C. Rigel Woodside(now U.S. Pat. No. 10,514,413), which claims benefit of U.S. provisionalApp. No. 62/400,018 entitled “Estimation of arc location in threedimensions” filed Sep. 26, 2016 in the names Matthew A. Cibula, Paul E.King, and C. Rigel Woodside. All of said applications are herebyincorporated by reference as if fully set forth herein.

FIELD OF THE INVENTION

The field of the present invention relates to electric arcs. Inparticular, apparatus and methods are disclosed for estimating thelocation of an electric arc in three dimensions.

SUMMARY

An inventive apparatus comprises a set of multiple magnetic fieldsensors, a data acquisition system operatively coupled to the magneticfield sensors, and a computer system operatively coupled to the dataacquisition system. The multiple magnetic field sensors are arrangedaround a lateral periphery of a current-containing volume into which aninput current flows and through which a primary electric current flowsin a predominantly longitudinal direction. The primary current flowsthrough first and second longitudinal electrical conductors positionedend-to-end within the current-containing volume and separated by an arcgap, and flows as one or more primary electric arcs spanning the arc gapand movable in two transverse dimensions within the arc gap between thefirst and second conductors. Each sensor of the set is positioned at acorresponding one of multiple distinct sensor positions; the sensorpositions are arranged among two or more distinct longitudinal positionsalong the current-containing volume and among two or more distinctcircumferential positions around the lateral periphery of thecurrent-containing volume. Each sensor of the set is arranged so as tomeasure magnetic field components in two or more spatial dimensions;each sensor of the set is characterized by one or more correspondingcalibration parameters. The data acquisition system is structured andconnected so as to convey to the computer system signals from themultiple sensors indicative of the measured magnetic field components.The computer system (comprising one or more electronic processors andone or more digital storage media coupled thereto) is structured,connected, and programmed so as to calculate an estimated transverseposition of the one or more primary electric arcs within the arc gap.The calculation is based at least in part on an estimated longitudinalposition of the arc gap, and two or more of the measured magnetic fieldcomponents along with one or more corresponding sensor positions orcalibration parameters. The current-containing volume can be enclosedwithin an electrically conductive chamber that defines the lateralperiphery of the current-containing volume, and the multiple sensorpositions can be located outside the chamber. The chamber can comprisean electric arc furnace, the first conductor can comprise an electrodeof the furnace, the second conductor can comprise an ingot formed withinthe furnace, and the furnace can be arranged so that the arc gap moveslongitudinally through the furnace as the input current flows during amelt period, causing the electrode to melt and shrink and the ingot togrow.

An inventive method, for estimating the transverse position of the oneor more primary electric arcs as a function of longitudinal position ofthe arc gap within the electric arc furnace during the melt period,comprises: (A) during the melt period, measuring the magnetic fieldcomponents in two or more spatial dimensions using the set of multiplemagnetic field sensors; (B) using the data acquisition system, conveyingfrom the multiple sensors to the computer system the signals from themultiple sensors indicative of the corresponding measured magnetic fieldcomponents; and (C) using the computer system, for each one of multiplemelt times within the melt period, calculating the correspondingestimated transverse position of the one or more primary electric arcswithin the arc gap. The calculation is based at least in part on thelongitudinal position of the arc gap at the corresponding melt time, andtwo or more of the magnetic field components, measured at thecorresponding melt time, along with one or more corresponding sensorpositions or calibration parameters.

Objects and advantages pertaining to locating an electric arc may becomeapparent upon referring to the example embodiments illustrated in thedrawings and disclosed in the following written description or appendedclaims, and shall fall within the scope of the present disclosure orappended claims.

This Summary is provided to introduce a selection of concepts in asimplified form that are further described below in the DetailedDescription. This Summary is not intended to identify key features oressential features of the claimed subject matter, nor is it intended tobe used as an aid in determining the scope of the claimed subjectmatter.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a schematic perspective view of an example inventivearrangement of magnetic sensors coupled to a computer system by a dataacquisition system.

FIG. 2 is a schematic longitudinal cross section of an arc furnace withan example inventive arrangement of magnetic sensors, and with a primaryelectric current flowing as a primary electric arc.

FIG. 3 is a schematic longitudinal cross section of the arc furnace withthe example inventive arrangement of magnetic sensors, and with aprimary electric current flowing as a primary electric arc and asecondary electric current flowing as a secondary electric arc.

FIG. 4 includes several graphs of dependence on longitudinal position ofcalculated magnetic field components arising from a simulated electricarc within a simulated non-coaxial arc furnace.

FIGS. 5A and 5B are schematic contour plots of the magnetic fieldmagnitude in an arc furnace with a longitudinal primary arc (FIG. 5A)and a transverse secondary arc (FIG. 5B).

FIGS. 6A and 6B are schematic contour plots of the magnetic fieldlongitudinal component in an arc furnace with a longitudinal primary arc(FIG. 6A) and a transverse secondary arc (FIG. 6B).

FIG. 7A shows an example of a measured distribution of estimatedtransverse arc positions along the length of an ingot. FIG. 7B showexamples of measured distributions of transverse arc position across aningot averaged over 1 second, 10 seconds, and 100 seconds, respectively.

FIGS. 8A, 8B, and 8C are schematic plots of measured and calculatedmagnetic field components as a function of arc gap longitudinal positionfor three different sensor rings.

FIGS. 9A and 9B are schematic plots of measured magnetic fieldlongitudinal components as a function of arc gap longitudinal positionfor three different senor rings in two different arc furnaces.

The embodiments depicted are shown only schematically: all features maynot be shown in full detail or in proper proportion, certain features orstructures may be exaggerated relative to others for clarity, and thedrawings should not be regarded as being to scale. The embodiments shownare only examples: they should not be construed as limiting the scope ofthe present disclosure or appended claims.

DETAILED DESCRIPTION OF EMBODIMENTS

The subject matter disclosed herein may be related to that disclosed inU.S. Pat. No. 8,111,059 entitled “Electric current locator” issued Feb.7, 2012 to King et al (hereinafter referred to as the '059 patent), saidpatent being incorporated by reference as if fully set forth herein.

An example of an inventive apparatus for estimating the location of anelectric arc comprises a set of multiple magnetic field sensors 200, adata acquisition system 298 operatively coupled to the magnetic fieldsensors 200, and a computer system 299 operatively coupled to the dataacquisition system (e.g., as in FIG. 1). In some instances disclosedherein the magnetic field sensors are referred to collectively orgenerically in the description, and indicated collectively orgenerically in the drawings, using the single reference number 200; inother instances herein the magnetic field sensors are referred toindividually or as subsets in the description, and indicatedindividually or as subsets in the drawings, using references numbers 200a, 200 b, 200 c, and so on. FIGS. 1 through 3 illustrate schematicallyexample arrangements of the multiple magnetic field sensors 200 around alateral periphery of a current-containing volume 10. In some examples(e.g., as in FIGS. 2 and 3) the current-containing volume 10 is theinterior volume of a chamber such as an electric arc furnace 100 and isbounded by its walls, which define the lateral periphery of thecurrent-containing volume 10. “Around the lateral periphery” can denotesensor positions exactly on the lateral boundary of thecurrent-containing volume, or sensor positions inside or outside thatlateral boundary, provided that the sensor positions span multipledistinct circumferential positions relative to the current-containingvolume 10. In many examples (e.g., as in FIGS. 2 and 3) the walls of thearc furnace 100 include an inner, electrically conductive crucible 101(often made of copper), an outer wall 102, and a cooling water jacket103 between them; in such examples the sensors 200 typically arepositioned on the outer surface of the outer wall 102. Although suchexamples are the focus of the present disclosure, the apparatus andmethods disclosed herein also can be employed for locating an electricarc within other types or arrangements of a current-carrying volume 10,while nevertheless remaining within the scope of the present disclosureor appended claims.

An input electric current 20 flows into the current-carrying volume 10;a portion of that input current flows within the current-carrying volume10 as a primary electric current 22. In some instances the input current20 and the primary current 22 are equal, i.e., all of the input current20 flows within the current-carrying volume 10 as the primary current 22(e.g., as in FIG. 2). In other instances the primary current 22 is lessthan the input current 20 (e.g., as in FIG. 3; discussed further below).The terms “longitudinal” and “transverse” as used herein are definedrelative to the desired direction of flow of the primary current 22. Aboundary of the current-containing volume 10 in a transverse directionis referred to as a lateral boundary or a lateral periphery. Forexamples in which the current-containing volume 10 is contained withinan electric arc furnace 100, the long axis of the furnace 100 definesthe longitudinal direction or dimension (e.g., vertical in FIGS. 2 and3), and directions perpendicular to that furnace axis are the transversedirections or dimensions (e.g., horizontal in FIGS. 2 and 3); side wallsof the arc furnace 100 define the lateral periphery of thecurrent-containing volume 10 within the arc furnace 100. Thelongitudinal direction or dimension may in some instances be referred toas the z-direction or z-dimension, or as the vertical direction ordimension; in such instances the transverse directions or dimensions maybe referred to as the x- and y-directions or x- and y-dimensions, or asthe horizontal directions or dimensions. Those additional designationsor descriptors are arbitrary, are made only for convenience ofdescription, and should not be construed as limiting the scope of thepresent disclosure or appended claims.

First and second longitudinal electrical conductors 110 and 120,respectively, are positioned end-to-end within the current-containingvolume 10 and are separated by an arc gap 115. In examples wherein thecurrent-containing volume 10 is the interior of the arc furnace 100, thefirst longitudinal electrical conductor 110 can comprise the electrode110 of the furnace 100, and the second longitudinal electrical conductor120 can comprise the ingot 120 formed within the furnace 100. Duringoperation of the furnace 100, the ingot 120 typically includes a pool ofmolten metal (i.e., the so-called melt pool 122) at its top surface(with the furnace 100 operated in its typical orientation, with its longaxis oriented substantially vertically and with the electrode 110positioned above the ingot 120). The primary current 22 flows throughthe first conductor 110, flows between the conductors 110/120 as one ormore primary electric arcs 30 that span the arc gap 115, and flowsthrough the second conductor 120. In the arc furnace example, theprimary current 22 flows through the electrode 110, flows between theelectrode 110 and the ingot 120 as one or more primary arcs 30 spanningthe arc gap 115, and flows through at least a portion of the ingot 120.In the arc furnace example, the primary current 22 flows from the ingot120 into the side walls of the crucible 101 and flows through at leastportions of the crucible walls as a return current 24. It is typicallyassumed that the primary current 22 flows primarily from near the top ofthe ingot 120 into the side walls of the crucible 101 before flowing asthe return current 24 (as in the drawings); it is also possible that atleast a portion of the primary current 22 flows into the crucible wallselsewhere along the ingot 120, or at the bottom of the ingot 120, beforeflowing as the return current 24. The scope of the present disclosureand appended claims shall encompass each of those alternatives as wellas any combination of one or more of those alternatives.

An electric arc furnace often is operated for the purpose of vacuum arcremelting, in which the electrode 110 is made of a high-value metal oralloy, and it is desired to improve the quality of the material (e.g.,by improved homogeneity of macroscopic or microscopic structure orcomposition, by reduction of impurities, and so forth). Under vacuumconditions (e.g., less than about 1 mmHg or less than about 0.1 mmHg), alarge electric current (e.g., several kiloamperes) is driven through theelectrode 110 to strike an arc 30 against a small amount of seedmaterial at the bottom of the crucible 101. The primary current 22flowing as the primary electric arc 30 causes the electrode 110 to meltat the arc gap 115 and form a melt pool 122. Solidification of themelted material forms and grows the ingot 120. As the remelting processproceeds, the ingot 120 grows, the electrode 110 shrinks, and the arcgap 115 moves upward through the arc furnace 100. The water jacket 103around the crucible 101 is employed to control the solidification rateand conditions to yield the desired properties of the ingot material.The electrode 110 has a smaller diameter then the crucible 101 (andhence the ingot 120 that is formed) to avoid current flow from theelectrode 110 to the walls of the crucible 101. Because of this, theelectrode 110 must be moved downward at the correct rate duringoperation of the furnace in order to maintain the correct height of thearc gap 115 (i.e., the distance between the electrode 110 and the ingot120) as the electrode 110 melts away and the ingot 120 grows. An arcfurnace 100 can be any suitable or desirable length; many examples areover 100 inches long, or over 200 inches long, or even longer.

The diameter of the electrode 110 typically is larger (usually muchlarger) than the transverse extent of the primary electric arc 30,thereby allowing the primary arc 30 to move in two transverse dimensionswithin the arc gap 115 between the electrode 110 and the ingot 120.Typical transverse dimensions of electrode 110 (e.g., the diameter of acylindrical electrode 110) range from about 12 inches to about 36 inchesor more; crucible transverse dimensions typically are about 2 to 4inches larger than the electrode dimensions. The primary arc 30, on theother hand, typically is no more than a few millimeters wide, allowingthe primary arc 30 to move relatively freely across the transversedimensions of the arc gap 115. That transverse movement of the primaryarc 30 can affect the quality of the material forming the ingot 120. Itwould be desirable to provide an estimate of the transverse position ofthe primary electric arc 30, as a function of longitudinal positionalong the ingot 120 (or equivalently, as a function of longitudinalposition within the furnace 100 of the arc gap 115). Such an estimate ofthe primary arc's position can take a variety of forms and can be usedfor a variety of purposes. In some examples, a detailed trajectory ofestimated positions of the primary arc 30 can be generated and stored;in other examples a distribution function can be generated that reflectsthe relative probability density that the primary arc 30 was at aparticular transverse position; that density can be averaged over anysuitable or desirable timescale, e.g., over 1 second, 10 seconds, 100seconds, or other suitable interval. The estimated arc transverseposition can be generated in real time as the furnace is operated, orcan be generated later in so-called offline processing.

The set of multiple magnetic sensors 200 measure magnetic fieldcomponents around the current-containing volume 10; those measuredvalues are used to calculate an estimated position of the primary arc30. Each magnetic sensor 200 is positioned at a corresponding one ofmultiple distinct sensor positions. The sensor positions are arrangedamong two or more distinct longitudinal positions along thecurrent-containing volume and among two or more distinct circumferentialpositions around the lateral periphery of the current-containing volume10 (e.g., as in FIGS. 1 through 3). This is in contrast to thearrangement of magnetic sensors disclosed in the '059 patent, in whichall the magnetic sensors are arranged at a single longitudinal positionalong the current-containing volume 10. Each magnetic sensor 200 isarranged so as to measure magnetic field components in two or morespatial dimensions; in many examples the magnetic sensors measuremagnetic field components in all three spatial dimensions. Each magneticsensor 200 is characterized by one or more corresponding calibrationparameters (discussed further below). The magnetic sensors 200 can be ofany suitable type, e.g., Hall effect sensors or magnetoresistivesensors. Discrete devices can be used to measure each spatial componentat a given sensor location, or multiple devices can be integrated into asingle, monolithic sensor that measures two or three magnetic fieldcomponents. In some examples, a single, integrated, 3D Hall effectsensor is located at each sensor position to measure the magnetic fieldcomponents in all three spatial dimensions. The output of such sensorstypically is a voltage proportional to the measured magnetic fieldcomponent. In one specific example, sensors 200 can measure up to about100 gauss with resolution of about 0.01 gauss.

The data acquisition system 298 can be of any suitable type orarrangement, and typically includes one or more analog-to-digitalconverters (A/Ds). The computer system 299 comprises one or moreelectronic processors and one or more digital storage media coupled tothe one or more processors. The data acquisition system 298 isstructured and connected so as to convey to the computer system 299signals from the multiple sensors 200 indicative of the measuredmagnetic field components. In some examples, the data acquisition systemcan be a separate set of one or more components or modules coupling thesensors 200 to the computer system 299. In some examples, the dataacquisition system 298 can be distributed among and integrated with thesensors 200 (e.g., by integration of one or more Hall sensors and one ormore A/Ds on a single chip or in a single device at each sensorposition). In some examples the data acquisition system can beintegrated with the computer system 299 (e.g., by integration of one ormore A/Ds into the computer system along with one or more processors andone or more storage media). All of those arrangements, and combinationsthereof, fall within the scope of the present disclosure or appendedclaims.

The computer system 299 is structured, connected, and programmed so asto calculate an estimated transverse position of the one or more primaryelectric arcs 30 within the arc gap 115. That calculation can beperformed in any suitable way using any suitable computationalalgorithm, modeling, or technique. The calculation of the estimatedprimary arc position is based at least in part on (i) longitudinalposition of the arc gap, and (ii) two or more of the measured magneticfield components along with the corresponding sensor position(s) orcalibration parameter(s). In some examples, the calculation is alsobased at least in part on the magnitude of the input electric current orthe primary electric current. In some examples, the longitudinalposition of the arc gap 115 is a known input parameter; in someexamples, the longitudinal position of the arc gap 115 is estimatedbased at least in part on one or more operational parameters, e.g.,electrode weight, elapsed time of operation of the arc furnace (i.e.,melt time), current or magnetic field fluctuations arising fromelectrode weldments, and so forth; in some examples, the longitudinalposition of the arc gap 115 is estimated based at least in part on themagnetic field components measured by the sensors 200. In some examples,measured values of the multiple magnetic field components measured bysome or all of the magnetic field sensors 200 are included in a set ofequations that also includes the position of the arc gap 115 (known,measured, or estimated) and the calibration parameter(s) or sensorposition for each corresponding sensor; the set of equations can alsoinclude other known, measured, or estimated operational or structuralparameters noted herein (e.g., current magnitude, furnace dimensions,and so forth). In examples wherein measured field values from some, butnot all, sensors 200 are employed in a given calculation, thecorresponding subset of sensors 200 employed to calculate a givenestimated position can vary according to, e.g., the correspondingestimated longitudinal position of the arc gap 115. In some examples,sensor positions, operational parameters (e.g., input current), orstructural parameters (e.g., furnace diameter or height, electrodediameter) can appear explicitly in the set of equations; in someexamples, one or more such quantities do not appear explicitly, butappear implicitly through their influence on the calibration parameters(e.g., a given sensor can have calibration parameters that varydepending on the size and shape of the furnace, or depending on itsposition on a given furnace). For purposes of the present disclosure, acalculation is “based at least in part” on a given quantity regardlessof whether that quantity influences the calculation explicitly orimplicitly. In some examples the set of equations includes only linearequations; in other examples the set of equations can include one ormore nonlinear equations. One specific example illustrating how such acalculation can be performed is disclosed in the '059 patent (albeit fora ring of sensors at only a single longitudinal position).

In some examples, the one or more corresponding calibration parametersfor each of the sensors 200 are derived from calculations ofposition-dependent magnetic fields arising from simulated electriccurrents within a simulated current-containing volume. In some examples,multiple simulations are run; in each run of the simulation a singlesimulated primary electric arc is placed at a corresponding transverseposition within a simulated arc gap at a corresponding longitudinalposition within the simulated current-containing volume, and thesimulated primary current flows through simulated first and secondconductors and the simulated primary arc. In some examples a simulatedreturn current flows in the opposite direction along the lateralperiphery of the current-containing volume (e.g., to simulate the returncurrent flowing along the lateral walls of the crucible 101). For eachrun of the simulation, the resulting magnetic field components arecalculated as a function of position at least along the lateralperiphery of the simulated current-containing volume (and within itsinterior as well, if needed or desired), using Maxwell's equations orany suitable subset or approximation thereof (e.g., Ampere's Law or theBiot-Savart Law) and using any suitable computational technique (e.g.,the finite element method). Multiple runs of the simulation areperformed, with the simulated primary arc located at correspondingdifferent transverse positions, and with the simulated arc gap locatedat corresponding different longitudinal positions. In one specificexample, magnetic field components are calculated for 27 differentprimary arc positions at each of 30 different longitudinal positions ofthe arc gap, for a total of 810 runs of the simulation. In examples of aso-called coaxial furnace, in which the input current 20 flows through avertical conductor into the top of the electrode 110, certainsimplifying approximations can be made in the simulations, based onassumed symmetries of the furnace 100. In so-called non-coaxialexamples, wherein, e.g., as in FIGS. 2 and 3, the input current 20 flowsinto the electrode 110 through a transverse conductor 104, a full 3Dcalculation typically must be performed in the simulations.

With the calculated field values in hand, multiple sensor positions areselected arranged around the lateral periphery of the current-containingvolume. The multiple sensor locations are arranged among two or moredistinct longitudinal positions and among two or more distinctcircumferential positions around the lateral periphery of thecurrent-containing volume. In some examples, the sensor positions arearranged in two or more rings located at corresponding distinctlongitudinal positions, with each ring including multiple sensorpositions at a single longitudinal position but at correspondingdistinct circumferential positions. In the examples of FIGS. 1 through3, three rings of sensors 200 a, 200 b, and 200 c are employed. In FIG.1 each of the three rings includes four sensors 200 a, 200 b, or 200 c;in the cross-sections of FIGS. 2 and 3 only two sensors 200 a, 200 b, or200 c of each ring are visible. Note that the “rings” do not necessarilycorrespond to an actual mechanical ring structure, but only to thegeometric arrangement of multiple sensor positions that make up the ring(i.e., at the single longitudinal position but multiple distinctcircumferential positions). Note that “at a single longitudinalposition” can denote a subset of the sensors 200 that are atsubstantially the same longitudinal position, or in some instances candenote a subset of the sensors 200 that fall within a sufficientlynarrow range of longitudinal positions to enable their respectivemeasured field components to be used together in certain calculationsthat depend on longitudinal position (discussed further below). Inaddition to the multiple rings, additional sensor positions can belocated at longitudinal positions that differ from any of the rings,e.g., a longitudinal line of 5 sensors 200 d through 200 h is employedin addition to three rings 200 a through 200 c in the examples of FIGS.1 through 3. Other numbers or arrangements of the non-ring sensors canbe employed. More generally, the sensors 200 need not be arranged in anyrings or longitudinal lines, provided that sensors 200 are located atmultiple distinct longitudinal positions and multiple distinctcircumferential positions, and those positions are known accurately.Other examples of suitable arrangements can include, e.g., one or morespirals, or a random scattering of sensors.

The simulations (however they are performed) yield magnetic fieldcomponents expected to be measured at a given sensor location for aprimary electric arc located at any one of the many combinations oftransverse arc position and longitudinal arc gap position used in thesimulations. From those calculated field components, one or morecalibration parameters can be determined for each sensor 200; in manyexamples, the one or more calibration parameters of a given magneticsensor 200 will vary with respect to the longitudinal position of thearc gap. The number and type of calibration parameters required tocharacterize each or the sensors 200 depends on the nature of themathematical model used in the simulations or to preform subsequentcalculations using those parameters. Within the scope of the presentdisclosure or appended claims, myriad measurement and computationschemes can be devised and implemented for generating suitablecalibration parameters for the magnetic sensors 200 and for estimatingthe position of an electric arc using those parameters in conjunctionwith measured magnetic field components. Some examples are describedbelow.

In some examples, a set of linear equations can be constructed from thesimulation data, in which each simulated current (at a known transverseposition and at a known arc gap longitudinal position) is expressed as alinear combination of a subset of calculated magnetic field components,each multiplied by a corresponding yet-to-be-determined calibrationparameter. In some examples, all field components calculated at allsensor positions can be included in each equation, In some examples,each equation can include only a corresponding subset of less than allof those calculated field components; the included field components canbe selected for each equation on any suitable basis, e.g., proximity ofthe corresponding sensor position to the simulated arc gap longitudinalposition. In some examples, the set of equations can be solved todetermine the calibration parameters.

To estimate a location of a real primary electric arc 30 at an unknownposition within a real arc furnace 100, those same equations can beused, with the now-determined calibration parameters and a set ofmagnetic field components measured by the real sensors 200, to calculatea distribution of current magnitudes among the simulated arc positions.That raw distribution can be used as an estimate of the primary arcposition, e.g., as a probability density distribution. Alternatively, acentroid of that raw current distribution can be calculated and used asan estimated position of the primary electric arc 30. In some instances,magnetic field components and calibration parameters of all sensors 200can be used together and used to estimate the primary arc transverseposition, and perhaps also the arc gap longitudinal position (if it isnot known or estimated by other means). In other instances, only asubset of the sensors are employed together for a given calculation(e.g., for a given estimated longitudinal position of the arc gap, onlysensors within a certain longitudinal distance of the arc gap might beemployed).

The calibration parameters can be calculated so as to compensate forother magnetic fields that might affect the magnetic field componentsmeasured by the sensors 200. For example, an arc furnace 100 typicallyincludes external current pathways that carry the input electric current20 to the electrode 110 or that carry the return current 24 from thefurnace walls (e.g., the transverse conductors 104 and 106 shown inFIGS. 2 and 3, carrying the input and return currents 20 and 24,respectively). Electric currents carried by those elements producemagnetic fields that are detected and measured by the sensors 200, andthose additional measured fields can introduce error into the estimationof the transverse position of the primary arc 30 in a non-coaxialfurnace. FIG. 4 shows an example of the variation in magnetic fieldcomponents, as a function of longitudinal position, that can arise in anon-coaxial furnace arrangement. To remove that source of error, thesimulations described above can include suitably arranged simulatedexternal current pathways, and the magnetic fields arising therefrom, sothat the calibration parameters generated from the simulation dataautomatically account for those additional fields. Similarly, thesimulations can include the earth's magnetic field, so that it also canbe accounted for in the calibration parameters calculated from thesimulation data. Note that the earth's magnetic field varies by locationon the surface of the earth; the simulations are run using the magneticfield value for the intended location of the arc furnace, and theresulting calibration parameters calculated from the simulation data arespecific to that furnace location. External magnetic fields arising fromother sources (e.g., from one or more adjacent arc furnaces) can besimilarly incorporated into simulations and thereby accounted for in theresulting calibration parameters. If an adjacent arc furnace issimilarly equipped with magnetic sensors, field components measured bythose sensors can be included in the calculated estimate of the primaryarc transverse position. If an adjacent arc furnace is sufficiently faraway, its current can be approximated as a simple line current and theresulting magnetic field components included in the calculation, e.g.,as a substantially uniform offset.

Calculation of sensor calibration parameters based on simulation datatypically assumes that the sensors 200 are perfectly aligned (e.g., thethree orthogonal measurement axes of a 3D sensor are perfectly alignedalong the longitudinal and transverse axes of the arc furnace). However,this is typically not the case in a real system of sensors 200positioned around an arc furnace 100; there is often some smalldeviation of the orientation of each sensor 200 from its idealizedorientation used in the simulation. Those deviations can be the same formultiple sensors (e.g., a framework or bracket holding multiple sensors200 is misaligned relative to the arc furnace 100) or can vary, oftensomewhat randomly, among the sensors (e.g., random misalignment ofindividual sensors 200 relative to the arc furnace 100); both of thosetypes of misalignment can occur simultaneously. In some examples,directional corrections can be incorporated into the calculation ofcalibration parameters. In some instances, those directional correctionscan be based on magnetic field components arising from a real testcurrent (not simulated) and measured by the sensors. In one arrangement,a straight conducting rod is placed within an arrangement of the sensors200 (e.g., with the sensors 200 mounted on a jig, bracket, framework, orother hardware that will be used for subsequent attachment of thesensors to the arc furnace 100, or with the sensors mounted on the arcfurnace 100 and with the conducting rods positioned within the arcfurnace 100), and a test current is run through the rod. Magnetic fieldcomponents measured by the mounted sensors 200 are compared to magneticfield components calculated in a simulation of the test current flowingthrough the rod among the sensors 200, and that comparison is used tocorrect the calibration parameters of the sensors 200.

As noted above, in some examples the multiple sensor positions includetwo or more rings of sensor positions, wherein each ring includesmultiple sensor positions arranged at substantially the samelongitudinal position along the current-containing volume and atmultiple distinct circumferential positions around the lateral peripheryof the current-containing volume. In some of those examples, themultiple sensor positions include three or more of the rings of sensorpositions (e.g., sensor rings 200 a, 200 b, and 200 c shown in FIGS.1-3). In some examples that include rings of sensor positions, thesensors 200 at the positions in the rings can be arranged so as tomeasure magnetic field components in at least both substantiallytransverse dimensions. Such an arrangement can be advantageouslyemployed in some examples for estimating the primary arc transverseposition, particularly when the arc gap 115 is relatively near thelongitudinal position of one of the rings, e.g., in furnaces of typicalsizes, within about two feet of a sensor ring 200 a/200 b/200 c in anon-coaxial furnace, or within about 8 feet of a sensor ring in acoaxial furnace (in each case, with suitable longitudinal variation ofthe calibration parameters according to the estimated longitudinalposition of the arc gap 115). In some examples that include two or morerings of sensor positions, two or more corresponding preliminaryestimates of the transverse position of the primary arc can becalculated, with each of those preliminary estimates being calculatedusing magnetic field components measured by sensors 200 of only one ofthe rings (or possibly including additional sensors 200 that are withina limited longitudinal distance of that ring). The estimated position ofthe primary arc 30 can then be calculated as a weighted average of thetwo or more preliminary estimates. Each preliminary estimate can beweighted according to its longitudinal distance from the arc gaplongitudinal position, with the weight factor of a given ring decreasingwith increasing longitudinal distance between that ring and the arc gapposition. A linear variation of the weight factor with longitudinaldistance can be employed, or any other suitable or desirablelongitudinal variation of the weight factors can be employed.

In one specific example method using the example arrangements of FIGS.1-3, a first preliminary estimated transverse position of the arc 30 canbe calculated using field values measured by only the sensor ring 200 a,a second preliminary estimated position calculated using only the sensorring 200 b, and a third preliminary estimated position calculated usingonly the sensor ring 200 c. For arc gap positions below the ring 200 a,only the first preliminary estimated position (calculated based on thesensor ring 200 a) is employed as the estimated arc position; for arcgap positions above the ring 200 c, only the third preliminary estimatedposition (calculated based on the sensor ring 200 c) is employed as theestimated arc position. For arc gap positions between the rings 200 aand 200 b, a suitably weighted average of the first and secondpreliminary estimated arc positons (calculated using the sensor rings200 a and 200 b, respectively) are employed to calculate the estimatedarc position; for arc gap positions between the rings 200 b and 200 c, asuitably weighted average of the second and third preliminary transversearc positons (calculated using the sensor rings 200 b and 200 c,respectively) are employed to calculate the estimated arc position.

In an electric arc furnace, it is desirable for the entire input current20 to flow through the conductors 110 and 120, and as the primary arc 30across the arc gap 115, as the primary current 22 (e.g., as in FIG. 2);in such instances the input current 20 and the primary current 22 aresubstantially equal. In some other instances however, one or moresecondary arcs 32 can appear, typically only transiently, that carry asecondary current 26, typically between the electrode 110 and the sidewall of the crucible 101 (e.g., as shown in FIG. 3; such a secondary arc32 is often referred to as a side arc). In such instances the inputcurrent 20 is substantially equal to a sum of the primary and secondarycurrents 22 and 26. Side arcing typically is regarded as undesirable inan arc furnace. First, it diverts a portion of the input current 20 awayfrom the primary arc 30 as the secondary current 26, reducing the amountof electrical energy delivered to the end of the electrode 110 to drivethe remelting process. Second, the side arc 32 can result in a distortedshape of the lower end or side surface of the electrode 110, which caninterfere with subsequent remelting. Third, a large flow of secondarycurrent 26 to the furnace wall via the side arc 32 results in localizedheating of a relatively small area of the crucible wall, which canintroduce crucible material (often copper) as an impurity into the ingot120, or can create a safety hazard by creating a “hot spot” on thecrucible wall, perhaps even leading to failure of the crucible wall. Itwould be desirable to provide a way to detect, and perhaps also measureor localize, secondary arcing when it occurs.

To achieve that purpose, in some examples two or more of the multiplesensors 200 are arranged so as to measure a magnetic field component ina substantially longitudinal dimension (sometimes referred to as a“z-field” component), in addition to the two transverse magnetic fieldcomponents. The computer system 299 is structured, connected, andprogrammed so as to so as to recognize one or more sets of measuredmagnetic field components that are indicative of a secondary electriccurrent 26 flowing in a predominantly transverse direction as asecondary electric arc 32 between the first or second electrode and thechamber. The calculation is based at least in part on the longitudinalposition of the arc gap 115, and two or more of the measured magneticfield components along with the corresponding sensor position(s) orcalibration parameter(s).

In many instances, a primary arc 30 produces a spatial distribution ofmagnetic field magnitude or magnetic field longitudinal component (e.g.,as in FIGS. 5A and 6A, respectively) that differs qualitatively fromanalogous spatial distributions of magnitude or longitudinal componentarising from a side arc 32 (e.g., as in FIGS. 5B and 6B, respectively).In some examples, in the absence of a secondary arc 32, the magnitude orlongitudinal component of the magnetic field is nearly symmetric in thelongitudinal direction, centered about the arc gap 115. In other words,in the absence of a secondary arc 32, the magnitude or longitudinalcomponent of the magnetic field at a certain distance above the arc gap115 is substantially equal to the magnitude or longitudinal component atthat same distance below the arc gap 115 and at the same circumferentialposition. In such examples, if a secondary arc 32 is present, however,that longitudinal symmetry is distorted and the magnitudes of those twomeasured magnitudes or longitudinal field components can differ. Thatlongitudinal asymmetry of the magnitude or longitudinal field components(about the arc gap 115) is a recognizable signature of the presence of asecondary arc 32 between, e.g., the electrode 110 and the side wall ofthe crucible 101. A suitable threshold value can be selected for thatasymmetry as the criterion for estimating that a secondary arc 32 ispresent. Such a threshold can be an absolute value, or can be suitablynormalized or offset, e.g., with respect to a baseline established byfield values previously measured during operation of the arc furnace100. Longitudinal variation of one or more other measured fieldcomponents (e.g., transverse field components) can be employed ifsuitable or desirable.

In other examples, azimuthal variation (at a given longitudinalposition) of the magnitude or one or more components of the magneticfield can be analyzed to enable recognition of the presence of a sidearc 32. In some examples, azimuthal variation of magnetic fieldmagnitude, or specific components thereof, arising from a primary arc 30typically includes an approximately sinusoidal variation with a singlemaximum and a single minimum about the periphery of thecurrent-containing volume 10. Azimuthal variation, of magnetic fieldmagnitude or components, arising from a side arc 32 typically differsfrom that relatively simple quasi-sinusoidal variation. Observeddeviation from the simple variation can be an indication of the presenceof a side arc 32.

A variety of actions can be taken based on a determination that a sidearc 32 is present, e.g., simply noting the occurrence of the secondaryarc 32 in documentation associated with that particular ingot 120,altering the input current 20 so as to suppress the secondary arc,cutting off the input current 20 to shut down the remelting processentirely as a safety measure, or other suitable, desirable, or necessaryaction.

In some examples, the computer system 299 can be further structured,connected, and programmed so as to calculate an estimated magnitude ofthe secondary electric current or an estimated position of the secondaryelectric arc. Simulations, similar to those described above for theprimary electric arc 30, typically would be needed in order to providethe necessary calibration parameters for such an estimation of secondaryarc position or secondary current magnitude. Such a calculation can bebased at least in part on the magnitude of the input current, thelongitudinal position of the arc gap, and two or more of the measuredmagnetic field magnitude or components along with the correspondingsensor position(s) or calibration parameter(s), and in some examples canbe carried out in a manner similar to any of the examples describedabove.

As already noted, as the remelting process proceeds, the ingot 120grows, the electrode 110 shrinks, and the arc gap 115 moves upwardthrough the arc furnace 100. The water jacket 103 around the crucible101 is employed to control the solidification rate and conditions toyield the desired properties of the ingot material. The electrode 110has a smaller diameter than the crucible 101 (and hence the ingot 120that is formed) to avoid current flow from the electrode 110 to thewalls of the crucible 101; consequently, the electrode 110 must be moveddownward at the correct rate during operation of the furnace in order tomaintain the correct height of the arc gap 115 (i.e., the distancebetween the electrode 110 and the ingot 120) as the electrode 110 meltsaway and the ingot 120 grows. In some typical examples of an arcfurnace, the inner diameter of the crucible 101 is about 6 inches toabout 40 inches, which is also the approximate diameter of the ingot 120that is formed. As the ingot solidifies it contracts and pulls away fromthe furnace side walls leaving a small gap (e.g., ca. 0.05 inches to 0.3inches). As the remelting process proceeds, at relatively short portion(e.g., ca. 2 inches to 12 inches or more) at the top of the ingot 120remains in contact with the furnace side walls, and provides a path forreturn current 24. In some examples, a portion of the return current 24can flow through the bottom of the ingot 120 into the walls of thecrucible 101.

The longitudinal position of the arc gap 115 (equivalently, the heightof the ingot 120) changes monotonically as the remelting processproceeds (in the arc furnace example), and so is a monotonic function ofthe melt time (i.e., the elapsed time beginning with the initiation ofthe flow of the input current 20; the entire range of melt timesbeginning with initiation of the input current flow and ending withtermination of the input current flow is referred to herein as the meltperiod). It is important to have a sufficiently accurate estimate of thearc gap longitudinal position as a function of melt time so that theestimated transverse arc position (also a function of melt time) can becorrelated with the corresponding longitudinal arc gap position, and inturn correlated with any observed variation of material properties ofthe ingot 120 along its longitudinal dimension. In some examples, thearc gap longitudinal position is also an input for the calculation ofthe estimated transverse position of the primary arc 30. The computersystem 299 is structured, connected, and programmed in any suitable wayso as to calculate an estimated arc gap longitudinal position thatchanges with melt time as the input current 20 flows. In some examples,the arc gap longitudinal position is assumed to be a linear function ofmelt time, with the slope determined by melt rate (mass/time), furnaceinner diameter, and material density, and can be calculated on thatbasis or directly measured (e.g., if the rate of decrease of the weightof the electrode 110 is available as a function of melt time, such as inan arc furnace equipped with a load cell or other device for monitoringthe changing weight of the electrode 110). An estimated arc gaplongitudinal position is calculated based on the particular melt timeand the slope, and that value is correlated with the correspondingestimated transverse position of the arc calculated at that particularmelt time. In some examples, as described above, the arc gap estimatedlongitudinal position is used in the calculation of the arc estimatedtransverse position; in other examples, the arc transverse position canbe estimated without using the arc gap position in the calculation. Theterm “correlated” means that the arc gap longitudinal position (orequivalently, the position along the length of the ingot 120) at a givenmelt time is associated with the arc transverse position estimated atthat given melt time (whether or not the arc gap longitudinal positionwas used in the calculation of the arc transverse position). A datasetthat includes estimated transverse positions of the arc 30 (perhapsaveraged over a suitable time interval) over the entire length of theingot 120 (equivalently, over the entire melt time), or a portionthereof, can be generated and stored, and employed for quality orprocess control, or characterization of or documentation for the ingot120. In the example of FIG. 7A, a distribution of transverse arcposition (in two orthogonal transverse dimensions in the two panels) isplotted along the length of an ingot; a significant transverse deviationof the arc position can be seen about halfway along the length of theingot. In the example of FIG. 7B, a distribution of transverse arcposition is plotted along the two transverse dimensions averaged over 1second (left panel), 10 seconds (middle panel), and 100 seconds (rightpanel).

In some examples, the magnetic field components measured by some or allof the sensors 200 can be employed to estimate the longitudinal positionof the arc gap 30 during the remelting process. For example, it has beenobserved in some examples (e.g., as in FIGS. 8A, 8B, and 8C) that theaverage magnetic field magnitude measured by a subset of the sensors 200at the same or nearby longitudinal positions (e.g., the sensors of onering) goes through a maximum with respect to the melt time as the arcgap 115 passes the longitudinal position of that subset of sensors 200.The respective longitudinal positions 201 a/201 b/201 c of the threerings of sensors 200 a/200 b/200 c are shown in the example of FIGS. 8A,8B, and 8C, along with calculated (202 a/202 b/202 c) and measured (203a/203 b/203 c) field values of the respective sensors 200 a (FIG. 8A),200 b (FIG. 8B), and 200 c (FIG. 8C). It has also been observed in someexamples (e.g., as in FIGS. 9A and 9B) that the average longitudinalcomponent of the magnetic field measured by such a subset of sensors 200goes through a minimum with respect to melt time as the arc gap 115passes the longitudinal position of the that sensor subset. Each suchlongitudinal subset of the sensors 200 can therefore act as acalibration point for estimating the arc gap longitudinal positionduring remelting process. As the remelting process progresses, thecomputer system 299 recognizes when the average magnetic field magnitudefor a particular longitudinal sensor subset goes through a maximum(either by reaching a predetermined value, or by reaching a relativemaximum value with respect to melt time and then beginning to decrease),or when the average longitudinal magnetic field component goes through aminimum. The longitudinal position that characterizes that longitudinalsubset is used to estimate the longitudinal position of the arc gap 115at the melt time when that maximum occurred. In some examples, thelongitudinal subset position is the estimated arc gap position; in otherexamples, the longitudinal sensor subset position is corrected by asuitable predetermined longitudinal offset, e.g., if it is observedduring a calibration procedure or simulation that the maximum or minimummeasured field magnitude or component occurs, not at the sensor subsetposition, but at some consistent longitudinal offset before or after(e.g., in the examples of FIGS. 9A and 9B, the minimum measured fieldcomponent occurs slightly before the arc gap 115 reaches thelongitudinal positions of the sensors rings 200 a/200 b/200 c). The arcgap longitudinal position can be estimated for one or more suchlongitudinal subsets of the sensors 200, and the estimated positions andcorresponding melt times can serve as fixed calibration points in thedependence of the arc gap longitudinal position with melt time.Intermediate arc gap positions can be interpolated between such fixedcalibration points, or between a fixed calibration point and a beginningor end point of the remelting process.

In other examples, the electrode 100 can provide one or more fixedcalibration points in the arc gap position versus melt time curve. Insome instances, an aggregated electrode 100 is fabricated by assemblingand welding together multiple pieces of the material to be remelted, andthe resulting weldments in the electrode 110 are at known positionsalong the electrode 110. It has been observed during remelting processesthat, upon the arc gap 115 reaching and passing such a weldment, arecognizable variation in the current, voltage, or one or more measuredmagnetic field components is often observed. If so, then the melt timeat which such a variation is recognized corresponds to the melt timewhen the arc gap passes the known weldment position, therebyestablishing a fixed calibration point. In a manner similar to thatdescribed above, one or more such fixed calibration points can beemployed to interpolate the entirearc-gap-longitudinal-position-versus-melt-time curve.

Portions of the systems and methods disclosed herein can be implementedas or with general or special purpose computers or servers or otherprogrammable hardware devices programmed through software, or ashardware or equipment “programmed” through hard wiring, or a combinationof the two. A “computer” or “server” can comprise a single machine orcan comprise multiple interacting machines (located at a single locationor at multiple remote locations). Computer programs or other softwarecode, if used, can be implemented in tangible, non-transient, temporaryor permanent storage or replaceable media, such as by includingprogramming in microcode, machine code, network-based or web-based ordistributed software modules that operate together, RAM, ROM, CD-ROM,CD-R, CD-R/W, DVD-ROM, DVD±R, DVD±R/W, hard drives, thumb drives, flashmemory, optical media, magnetic media, semiconductor media, or anyfuture computer-readable storage media.

Electronic indicia of any measured or calculated quantities, anycalibration or modeling parameters, any programming code orinstructions, and so forth can be read from, received from, written to,or stored on any of the tangible, non-transitory computer-readable mediamentioned herein.

In addition to the preceding, the following examples fall within thescope of the present disclosure or appended claims:

Example 1. An apparatus for estimating a location of an electric arc,the apparatus comprising a set of multiple magnetic field sensors, adata acquisition system operatively coupled to the magnetic fieldsensors, and a computer system operatively coupled to the dataacquisition system, wherein: (a) the multiple magnetic field sensors arearranged around a lateral periphery of a current-containing volume intowhich an input electric current flows and within which at least aportion of the input current flows, in a predominantly longitudinaldirection as a primary electric current, (i) through at least portionsof first and second longitudinal electrical conductors positionedend-to-end within the current-containing volume and separated by an arcgap, and (ii) as one or more primary electric arcs spanning the arc gapand movable in two transverse dimensions within the arc gap between thefirst and second conductors; (b) each sensor of the set is positioned ata corresponding one of multiple distinct sensor positions, and thesensor positions are arranged among two or more distinct longitudinalpositions along the current-containing volume and among two or moredistinct circumferential positions around the lateral periphery of thecurrent-containing volume; (c) each sensor of the set is arranged so asto measure magnetic field components in two or more spatial dimensionsand is characterized by one or more corresponding calibrationparameters; (d) the data acquisition system is structured and connectedso as to convey to the computer system signals from the multiple sensorsindicative of the corresponding measured magnetic field components; and(e) the computer system comprises one or more electronic processors andone or more digital storage media coupled thereto, and is structured,connected, and programmed so as to calculate an estimated transverseposition of the one or more primary electric arcs within the arc gap,that calculation being based at least in part on longitudinal positionof the arc gap, and two or more of the measured magnetic fieldcomponents along with one or more corresponding sensor positions orcalibration parameters.

Example 2. The apparatus of Example 1 wherein each sensor of the set isarranged so as to measure magnetic field components in three spatialdimensions.

Example 3. The apparatus of any one of Examples 1 or 2 wherein thecalculation is based at least in part on magnitude of the input electriccurrent or the primary electric current.

Example 4. The apparatus of any one of Examples 1 through 3 wherein theone or more corresponding calibration parameters for each sensor arederived from calculations, at multiple distinct transverse positions andat multiple distinct longitudinal positions of a simulated arc gap, ofmagnetic field components at the corresponding sensor position arisingfrom a simulated input current flowing in a single simulated primaryelectrical arc, so that for one or more of the multiple sensors thecorresponding one or more calibration parameters vary with thelongitudinal positon of the arc gap.

Example 5. The apparatus of any one of Examples 1 through 4 wherein oneor more calculations include corrections for magnetic field componentsarising from external conductors carrying the primary current into thecurrent-containing volume or carrying a return current out of thecurrent-containing volume.

Example 6. The apparatus of any one of Examples 1 through 5 wherein oneor more calculations include measured directional corrections to theorientations of the measured magnetic field components for each sensor.

Example 7. The apparatus of any one of Examples 1 through 6 wherein oneor more calculations include measured corrections arising from externalmagnetic fields in which the current-carrying volume is immersed.

Example 8. The apparatus of any one of Examples 1 through 7 wherein themultiple sensor positions include two or more rings of sensor positions,wherein each ring includes multiple sensor positions arranged atsubstantially the same longitudinal position along thecurrent-containing volume and at multiple distinct circumferentialpositions around the lateral periphery of the current-containing volume.

Example 9. The apparatus of Example 8 wherein the multiple sensorpositions include three or more of the rings of sensor positions.

Example 10. The apparatus of any one of Examples 8 or 9 wherein thosesensors positioned at corresponding sensor positions of the two or morerings are arranged so as to measure magnetic field components in twosubstantially transverse dimensions.

Example 11. The apparatus of any one of Examples 8 through 10 whereinthe multiple sensor positions include one or more sensor positions thatare positioned at corresponding longitudinal positions along thecurrent-containing volume that differ from the correspondinglongitudinal positions of the two or more rings of sensor positions.

Example 12. The apparatus of any one of Examples 8 through 11 whereinthe estimated transverse position of the one or more primary electricarcs is a weighted average of estimated arc transverse positionscalculated for each ring using measured field components from sensors atcorresponding sensor positions of only that ring, and the estimated arctransverse positions for each ring are weighted according to acorresponding longitudinal distance between that ring and the estimatedlongitudinal position of the arc gap, with a corresponding weight factordecreasing with increasing distance between the corresponding ring andthe estimated arc gap position.

Example 13. The apparatus of any one of Examples 1 through 12 whereinthe current-containing volume is enclosed within a chamber that definesthe lateral periphery of the current-containing volume, and the multiplesensor positions are located outside the chamber.

Example 14. The apparatus of Example 13 wherein two or more of themultiple sensors are arranged so as to measure a magnetic fieldcomponent in a substantially longitudinal dimension, and the computersystem is structured, connected, and programmed so as to recognize oneor more sets of measured magnetic field magnitudes or longitudinalcomponents that are indicative of a secondary electric current flowingin a predominantly transverse direction as a secondary electric arcbetween the first or second conductor and the chamber, that recognitionbeing based at least in part on the magnitude of the input current, anestimated longitudinal position of the arc gap, and two or more of themeasured longitudinal magnetic field components along with one or morecorresponding sensor positions or calibration parameters.

Example 15. The apparatus of Example 14 wherein the computer system isstructured, connected, and programmed so as to calculate an estimatedmagnitude of the secondary electric current or an estimated position ofthe secondary electric arc, that calculation being based at least inpart on the magnitude of the input current, an estimated longitudinalposition of the arc gap, and two or more of the measured longitudinalmagnetic field components along with one or more corresponding sensorpositions or calibration parameters.

Example 16. The apparatus of any one of Examples 14 or 15 wherein, withno secondary current flowing, the input current is substantially equalto the primary current.

Example 17. The apparatus of any one of Examples 14 or 15 wherein, withone or more secondary electric arcs present, the input current issubstantially equal to a sum of the primary and secondary currents.

Example 18. The apparatus of any one of Examples 13 through 17 furthercomprising the chamber, wherein the chamber comprises an electric arcfurnace, the first conductor comprises an electrode of the furnace, thesecond conductor comprises an ingot formed within the furnace, and thefurnace is arranged so that the arc gap moves longitudinally through thefurnace as the input current flows during a melt period, causing theelectrode to melt and shrink and the ingot to grow.

Example 19. The apparatus of Example 18 wherein the computer system isstructured, connected, and programmed so as to calculate an estimatedarc gap longitudinal position that changes with melt time as the inputcurrent flows during the melt period.

Example 20. The apparatus of Example 19 wherein the longitudinalposition of one or more selected sensors that register a maximummeasured magnetic field magnitude, with respect to melt time as theinput current flows during the melt period, is used to estimatelongitudinal position of the arc gap at the melt time when the selectedsensors register that maximum measured magnetic field magnitude.

Example 21. The apparatus of any one of Examples 19 or 20 wherein thelongitudinal position of one or more selected sensors that register aminimum measured magnetic field longitudinal component, with respect tomelt time as the input current flows during the melt period, is used toestimate longitudinal position of the arc gap at the melt time when theselected sensors register that minimum measured magnetic fieldlongitudinal component.

Example 22. The apparatus of any one of Examples 20 or 21 wherein theselected sensors are arranged in a ring around the periphery of thecurrent-carrying volume at a common longitudinal position.

Example 23. The apparatus of any one of Examples 19 through 22 wherein(i) the electrode includes one or more weldments at known longitudinalpositions that produce a recognizable alteration of one or more magneticfield components as the electrode melts and the arc gap passes eachweldment, (ii) the computer system is structured, connected, andprogrammed so as to recognize the alteration detected by one or more ofthe sensors, and (iii) a longitudinal position of corresponding weldmentof the electrode is used to estimate the longitudinal position of thearc gap at a melt time when the alteration occurs.

Example 24. The apparatus of any one of Examples 19 through 23 whereinthe computer system is structured, connected, and programmed so as tocalculate an estimated arc gap longitudinal position as a function oftime as an interpolation between arc gap positions estimated by weldmentposition, by minimum longitudinal field component detection, or bymaximum field magnitude detection.

Example 25. The apparatus of any one of Examples 19 through 24 whereinthe arc gap longitudinal position is estimated based at least in part onsize, shape, density, and weight of the first longitudinal electricalconductor, and duration and magnitude of the primary current flow.

Example 26. A method, using the apparatus of any one of Examples 1through 25, for estimating transverse position of one or more primaryelectric arcs as a function of longitudinal position of an arc gapwithin an electric arc furnace, during a melt period during which aninput electric current flows into the arc furnace and at least a portionof the input current flows, in a predominantly longitudinal direction asa primary electric current, (i) through at least portions of first andsecond longitudinal electrical conductors positioned end-to-end withinthe arc furnace and separated by the arc gap, and (ii) as the one ormore primary electric arcs spanning the arc gap and movable in twotransverse dimensions within the arc gap between the first and secondconductors, the method comprising: (A) during the melt period, measuringmagnetic field components in two or more spatial dimensions using a setof multiple magnetic field sensors, with each sensor positioned at acorresponding one of multiple sensor positions arranged around a lateralperiphery of the arc furnace, wherein the multiple sensor positions arearranged among two or more distinct longitudinal positions along the arcfurnace and among two or more distinct circumferential positions aroundthe lateral periphery of the arc furnace, and wherein each sensor ischaracterized by one or more corresponding calibration parameters; (B)using a data acquisition system structured and connected therefor,conveying from the multiple sensors to a computer system signals fromthe multiple sensors indicative of the corresponding measured magneticfield components; and (C) using the computer system that is structured,connected, and programmed therefor, for each one of multiple melt timeswithin the melt period, calculating a corresponding estimated transverseposition of the one or more primary electric arcs within the arc gap,that calculation being based at least in part on the longitudinalposition of the arc gap at the corresponding melt time, and two or moreof the magnetic field components, measured at the corresponding melttime, along with one or more corresponding sensor positions orcalibration parameters.

It is intended that equivalents of the disclosed example embodiments andmethods shall fall within the scope of the present disclosure orappended claims. It is intended that the disclosed example embodimentsand methods, and equivalents thereof, may be modified while remainingwithin the scope of the present disclosure or appended claims.

In the foregoing Detailed Description, various features may be groupedtogether in several example embodiments for the purpose of streamliningthe disclosure. This method of disclosure is not to be interpreted asreflecting an intention that any claimed embodiment requires morefeatures than are expressly recited in the corresponding claim. Rather,as the appended claims reflect, inventive subject matter may lie in lessthan all features of a single disclosed example embodiment. Thus, theappended claims are hereby incorporated into the Detailed Description,with each claim standing on its own as a separate disclosed embodiment.However, the present disclosure shall also be construed as implicitlydisclosing any embodiment having any suitable set of one or moredisclosed or claimed features (i.e., a set of features that are neitherincompatible nor mutually exclusive) that appear in the presentdisclosure or the appended claims, including those sets that may not beexplicitly disclosed herein. In addition, for purposes of disclosure,each of the appended dependent claims shall be construed as if writtenin multiple dependent form and dependent upon all preceding claims withwhich it is not inconsistent. It should be further noted that the scopeof the appended claims does not necessarily encompass the whole of thesubject matter disclosed herein.

For purposes of the present disclosure and appended claims, theconjunction “or” is to be construed inclusively (e.g., “a dog or a cat”would be interpreted as “a dog, or a cat, or both”; e.g., “a dog, a cat,or a mouse” would be interpreted as “a dog, or a cat, or a mouse, or anytwo, or all three”), unless: (i) it is explicitly stated otherwise,e.g., by use of “either . . . or,” “only one of,” or similar language;or (ii) two or more of the listed alternatives are mutually exclusivewithin the particular context, in which case “or” would encompass onlythose combinations involving non-mutually-exclusive alternatives. Forpurposes of the present disclosure and appended claims, the words“comprising,” “including,” “having,” and variants thereof, wherever theyappear, shall be construed as open ended terminology, with the samemeaning as if the phrase “at least” were appended after each instancethereof, unless explicitly stated otherwise. For purposes of the presentdisclosure or appended claims, when terms are employed such as “aboutequal to,” “substantially equal to,” “ca.”, “greater than about,” “lessthan about,” and so forth, in relation to a numerical quantity, standardconventions pertaining to measurement precision and significant digitsshall apply, unless a differing interpretation is explicitly set forth.For null quantities described by phrases such as “substantiallyprevented,” “substantially absent,” “substantially eliminated,” “aboutequal to zero,” “negligible,” and so forth, each such phrase shalldenote the case wherein the quantity in question has been reduced ordiminished to such an extent that, for practical purposes in the contextof the intended operation or use of the disclosed or claimed apparatusor method, the overall behavior or performance of the apparatus ormethod does not differ from that which would have occurred had the nullquantity in fact been completely removed, exactly equal to zero, orotherwise exactly nulled.

In the appended claims, any labelling of elements, steps, limitations,or other portions of a claim (e.g., first, second, etc., (a), (b), (c),etc., or (i), (ii), (iii), etc.) is only for purposes of clarity, andshall not be construed as implying any sort of ordering or precedence ofthe claim portions so labelled. If any such ordering or precedence isintended, it will be explicitly recited in the claim or, in someinstances, it will be implicit or inherent based on the specific contentof the claim. In the appended claims, if the provisions of 35 USC §112(f) are desired to be invoked in an apparatus claim, then the word“means” will appear in that apparatus claim. If those provisions aredesired to be invoked in a method claim, the words “a step for” willappear in that method claim. Conversely, if the words “means” or “a stepfor” do not appear in a claim, then the provisions of 35 USC § 112(f)are not intended to be invoked for that claim.

If any one or more disclosures are incorporated herein by reference andsuch incorporated disclosures conflict in part or whole with, or differin scope from, the present disclosure, then to the extent of conflict,broader disclosure, or broader definition of terms, the presentdisclosure controls. If such incorporated disclosures conflict in partor whole with one another, then to the extent of conflict, thelater-dated disclosure controls.

The Abstract is provided as required as an aid to those searching forspecific subject matter within the patent literature. However, theAbstract is not intended to imply that any elements, features, orlimitations recited therein are necessarily encompassed by anyparticular claim. The scope of subject matter encompassed by each claimshall be determined by the recitation of only that claim.

What is claimed is:
 1. A method comprising: (A) directing an inputelectric current to flow into a first longitudinal electrical conductorpositioned within a current-containing volume, so that at least aportion of the input electric current flows as a primary electriccurrent in a predominantly longitudinal direction (i) through at leastportions of the first conductor, (ii) through at least portions of asecond longitudinal electrical conductor positioned within thecurrent-containing volume end-to-end with respect to the firstconductor, the first and second conductors being separated by an arcgap, and (iii) as one or more primary electric arcs spanning the arc gapand movable in two transverse dimensions within the arc gap between thefirst and second conductors; (B) using a set of multiple magnetic fieldsensors, measuring magnetic field components in two or more spatialdimensions at two or more distinct longitudinal positions along alateral periphery of the current-containing volume and at two or moredistinct circumferential positions around the lateral periphery of thecurrent-containing volume; (C) using a data acquisition system,conveying from the multiple magnetic field sensors to a computer systemsignals indicative of the magnetic field components measured in part(B); and (D) using the computer system, calculating an estimatedtransverse position of the one or more primary electric arcs within thearc gap, that calculation being based at least in part on an estimatedlongitudinal position of the arc gap relative to the sensors and two ormore of the magnetic field components measured by the sensors at two ormore distinct longitudinal positions along the current-containing volumeand at two or more distinct circumferential positions around the lateralperiphery of the current-containing volume.
 2. The method of claim 1wherein each sensor of the set is arranged so as to measure magneticfield components in three spatial dimensions.
 3. The method of claim 1wherein the calculation is based at least in part on magnitude of theinput electric current or the primary electric current.
 4. The method ofclaim 1 wherein the calculation of part (D) is based in part on one ormore corresponding calibration parameters for each sensor that arederived from calculations, at multiple distinct transverse positions andat multiple distinct longitudinal positions of a simulated arc gap, ofmagnetic field components at the corresponding sensor position arisingfrom a simulated input current flowing in a single simulated primaryelectrical arc, so that for one or more of the multiple sensors thecorresponding one or more calibration parameters vary with thelongitudinal position of the arc gap.
 5. The method of claim 4 whereinone or more calculations include one or more of (i) corrections formagnetic field components arising from external conductors carrying theprimary current into the current-containing volume or carrying a returncurrent out of the current-containing volume, (ii) measured directionalcorrections to the orientations of the measured magnetic fieldcomponents for each sensor, or (iii) measured corrections arising fromexternal magnetic fields in which the current-carrying volume isimmersed.
 6. The method of claim 1 wherein the multiple sensors includetwo or more rings of sensors, wherein each ring includes multiplesensors arranged at substantially the same longitudinal position alongthe current-containing volume and at multiple distinct circumferentialpositions around the lateral periphery of the current-containing volume.7. The method of claim 6 wherein the multiple sensors include three ormore of the rings of sensors.
 8. The method of claim 6 wherein thosesensors of the two or more rings are arranged so as to measure magneticfield components in two substantially transverse dimensions.
 9. Themethod of claim 6 wherein the multiple sensors include one or moresensors that are positioned at corresponding longitudinal positionsalong the current-containing volume that differ from the correspondinglongitudinal positions of the two or more rings of sensors.
 10. Themethod of claim 6 wherein the estimated transverse position of the oneor more primary electric arcs is a weighted average of estimated arctransverse positions calculated for each ring using measured fieldcomponents from sensors of only that ring, and the estimated arctransverse positions for each ring are weighted according to acorresponding longitudinal distance between that ring and the estimatedlongitudinal position of the arc gap, with a corresponding weight factordecreasing with increasing distance between the corresponding ring andthe estimated arc gap position.
 11. The method of claim 1 wherein thecurrent-containing volume is enclosed within a chamber that defines thelateral periphery of the current-containing volume, and the multiplesensor positions are located outside the chamber.
 12. The method ofclaim 11 wherein two or more of the multiple sensors are arranged so asto measure a magnetic field component in a substantially longitudinaldimension, and the method further comprises, using the computer system,recognizing one or more sets of measured magnetic field magnitudes orlongitudinal components that are indicative of a secondary electriccurrent flowing in a predominantly transverse direction as a secondaryelectric arc between the first or second conductor and the chamber, thatrecognition being based at least in part on the magnitude of the inputcurrent, an estimated longitudinal position of the arc gap, and two ormore of the measured longitudinal magnetic field components.
 13. Themethod of claim 12, wherein the method further comprises calculating,using the computer system, an estimated magnitude of the secondaryelectric current or an estimated position of the secondary electric arc,that calculation being based at least in part on the magnitude of theinput current, an estimated longitudinal position of the arc gap, andtwo or more of the measured longitudinal magnetic field components. 14.The method of claim 12 wherein (i) with no secondary current flowing,the input current is substantially equal to the primary current, and(ii) with one or more secondary electric arcs present, the input currentis substantially equal to a sum of the primary and secondary currents.15. The method of claim 1, wherein: (E) the current-containing volume isthe interior of an electric arc furnace; (F) the first conductorincludes an electrode within the electric arc furnace; (G) the secondconductor includes an ingot formed within the electric arc furnace; (H)the input current flows and the magnetic field components are measuredduring a melt period during which the electrode melts and shrinks, theingot grows, and the arc gap moves longitudinally within the arcfurnace; (I) the method further comprises calculating the estimatedtransverse position of the one or more primary electric arcs at each oneof multiple melt times within the melt period.
 16. The method of claim15 further comprising, using the computer system, calculating anestimated arc gap longitudinal position that changes with melt time asthe input current flows during the melt period.
 17. The method of claim16 wherein the longitudinal position of one or more selected sensorsthat register a maximum measured magnetic field magnitude, with respectto melt time as the input current flows during the melt period, is usedto estimate longitudinal position of the arc gap at the melt time whenthe selected sensors register that maximum measured magnetic fieldmagnitude.
 18. The method of claim 16 wherein the longitudinal positionof one or more selected sensors that register a minimum measuredmagnetic field longitudinal component, with respect to melt time as theinput current flows during the melt period, is used to estimatelongitudinal position of the arc gap at the melt time when the selectedsensors register that minimum measured magnetic field longitudinalcomponent.
 19. The method of claim 16 wherein (i) the electrode includesone or more weldments at known longitudinal positions that produce arecognizable alteration of one or more magnetic field components as theelectrode melts and the arc gap passes each weldment, and (ii) themethod further comprises, using the computer system, recognizing thealteration detected by one or more of the sensors, and (iii) alongitudinal position of corresponding weldment of the electrode is usedto estimate the longitudinal position of the arc gap at a melt time whenthe alteration occurs.
 20. The method of claim 16 wherein the computersystem is structured, connected, and programmed so as to calculate anestimated arc gap longitudinal position as a function of time as aninterpolation between arc gap positions estimated by weldment position,by minimum longitudinal field component detection, or by maximum fieldmagnitude detection.
 21. The method of claim 16 wherein the arc gaplongitudinal position is estimated based at least in part on size,shape, density, and weight of the first longitudinal electricalconductor, and duration and magnitude of the primary current flow.